Prove that the following mappings are Isometries.

[Daron]

Homework Statement

Verify that the following mappings are isometries on R^2

Reflection Through the Origin
Translation
Rotation

Homework Equations

Qualities of a metric:

d(x,y) = d(y,x)
d(x,x) = 0
d(x,y) = 0 <=> x = y
d(x,y) =< d(x,z) +d(z,y)

The Attempt at a Solution

As a metric hasn't been specified, I have been trying to prove this for a general metric using just the intrinsic qualities. I haven't had much luck, though.
I know that all three are straightforward to prove in Euclidean Space, which gives a metric. But is there a simple proof for a general metric?

I may have misunderstood the meaning of Verify, but would nevertheless like a proof if there is one.

 

[LCKurtz]

It isn't true for general metrics. Think about the taxicab metric on R² and rotation.